Optimal. Leaf size=35 \[ \frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2454, 2389, 2295} \[ \frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {\operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+\frac {b}{x^2}\right )}{2 b}\\ &=\frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 0.97 \[ \frac {1}{2} \left (\frac {p}{x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 41, normalized size = 1.17 \[ \frac {b p - b \log \relax (c) - {\left (a p x^{2} + b p\right )} \log \left (\frac {a x^{2} + b}{x^{2}}\right )}{2 \, b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 65, normalized size = 1.86 \[ -\frac {a p \log \left (a x^{2} + b\right )}{2 \, b} + \frac {a p \log \relax (x)}{b} - \frac {p \log \left (a x^{2} + b\right )}{2 \, x^{2}} + \frac {p \log \left (x^{2}\right )}{2 \, x^{2}} + \frac {b p - b \log \relax (c)}{2 \, b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 1.43 \[ \frac {a p}{2 b}-\frac {a \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{2 b}+\frac {p}{2 x^{2}}-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 54, normalized size = 1.54 \[ -\frac {1}{2} \, b p {\left (\frac {a \log \left (a x^{2} + b\right )}{b^{2}} - \frac {a \log \left (x^{2}\right )}{b^{2}} - \frac {1}{b x^{2}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 47, normalized size = 1.34 \[ \frac {p}{2\,x^2}-\frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{2\,x^2}-\frac {a\,p\,\ln \left (a\,x^2+b\right )}{2\,b}+\frac {a\,p\,\ln \relax (x)}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.65, size = 58, normalized size = 1.66 \[ \begin {cases} - \frac {a p \log {\left (a + \frac {b}{x^{2}} \right )}}{2 b} - \frac {p \log {\left (a + \frac {b}{x^{2}} \right )}}{2 x^{2}} + \frac {p}{2 x^{2}} - \frac {\log {\relax (c )}}{2 x^{2}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (a^{p} c \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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